PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 98(112) (2015), 281–285 DOI: 10.2298/PIM1512281M

A CONVERGENCE THEOREM OF MULTI-STEP ITERATIVE SCHEME FOR NONLINEAR MAPS

Adesanmi Alao Mogbademu

Dedicated to Professor Z. Xue for his unique style of mentoring.

Abstract. Let K be a nonempty closed convex subset of a real Banach space X, T : K → K a nearly uniformly L-Lipschitzian (with sequence {r_{n} }) asymptotically generalized Φ-hemicontractive mapping (with sequence k_{n} ⊂ [1, ∞), lim_{n→∞} k_{n} = 1) such that F (T) = {ρ ∈ K : T ρ = ρ}. Let {α_{n} }_{n⩾0} , {β_{n}^{k}}_{n⩾0} be real sequences in [0, 1] satisfying the conditions:

(i) ∑ _{n⩾0} α_{n} = ∞

(ii) lim_{n→∞} α_{n} , β_{n}^{k} = 0, k = 1, 2, …, p − 1.

For arbitrary x_{0} ∈ K, let {x_{n} }_{n⩾0} be a multi-step sequence iteratively defined by

x_{n+1} = (1 − α_{n})x_{n} + α_{n} T^{n} y_{n}^{1} , n ⩾ 0,
y_{n}^{k} = (1 − β_{n}^{k})x_{n} + β_{n}^{k}T^{n} y_{n}^{k+1} , k = 1, 2, …, p − 2,
(0.1)  y_{n}^{p−1} = (1 − β_{n}^{p−1} )x_{n} + β_{n}^{p−1} T^{n} x_{n} , n ⩾ 0, p ⩾ 2.

Then, {x_{n} }_{n⩾0} converges strongly to ρ ∈ F (T). The result proved in this note significantly improve the results of Kim et al. [2].

1. Introduction

Let X be a real Banach space and J the normalized duality mapping from X into 2^{X^{∗}} defined by J(x) = {f ∈ X^{∗} : ⟨x, f⟩ = ∥x∥^{2} = ∥f∥^{2}}, where X^{∗} denotes the dual space of real Banach space X and ⟨., .⟩ denotes the generalized duality pairing between elements of X and X^{∗}. We first recall and define some concepts as follows. Let K be a nonempty subset of real Banach space X.

Definitions 1. Let T : K → K be a mapping.

(1) T is said to be uniformly L-Lipschitzian [1, 5] if there exists a constant L > 0 such that ∥T^{n}x − T^{n}y∥ ⩽ L∥x − y∥, for any x, y ∈ K and ∀n ⩾ 1.

2010 Mathematics Subject Classification: Primary 47H10; Secondary 46A03L. Key words and phrases: Mann iteration; multi-step iteration; asymptotically generalized Φ-

hemi contractive mappings; nearly Lipschitzian mapping; uniformly L-Lipschitzian; Banach space. Communicated by Stevan Pilipović.

(2) T is said to be asymptotically generalized Φ-hemicontractive with sequence {k_{n}}_{n⩾0} if F (T) ≠ ∅ and for each n ∈ N and x ∈ K, x^{∗} ∈ F (T), there exists constant k_{n} ⩾ 1 with lim_{n→∞} k_{n} = 1, strictly increasing function Φ : [0, ∞) → [0, ∞) with Φ(0) = 0 such that

⟨T^{n}x − T^{n}x^{∗}, j(x − x^{∗})⟩ ⩽ k_{n}∥x − x^{∗}∥^{2} − Φ(∥x − x^{∗}∥).

The class of asymptotically generalized Φ-hemicontractive mapping is the most general among those defined in [5].

(3) A mapping T : K → X is called Lipschitzian if there exists a constant L > 0 such that

∥T x − T y∥ ⩽ L∥x − y∥, for all x, y ∈ K

and is called generalized Lipschitzian if there exists a constant L > 0 such that

∥T x − T y∥ ⩽ L(∥x − y∥ + 1), for all x, y ∈ K.

It is obvious that the class of generalized Lipschitzian map includes the class of Lipschitz map. Sahu [5] introduced the following new class of nonlinear mappings which are more general than the class of generalized Lipschitzian mappings and the class of uniformly L-Lipschitzian mappings. Fix a sequence {r_{n}}_{n⩾0} in [0, ∞] with r_{n} → 0.

(4) A mapping T : K → K is called nearly Lipschitzian with respect to {r_{n}} if for each n ∈ N, there exists a constant k_{n} > 0 such that

∥T^{n}x − T^{n}y∥ ⩽ k_{n}(∥x − y∥ + r_{n}), for all x, y ∈ K.

A nearly Lipschitzian mapping T with sequence {r_{n}}_{n⩾0} is said to be nearly uniformly L-Lipschitzian if k_{n} = L for all n ∈ N.

Observe that the class of nearly uniformly L-Lipschitzian mapping is more general than the class of uniformly L-Lipschitzian mappings. We establish a strong convergence theorem for a more general class of map in real Banach space. It is worth noting that comparing [2, Theorem 2.1] our result have the following features:

(i) The modified Mann iterative process is replaced by Multi-step iterative process. (ii) We removed the condition that {r_{n} /α_{n} } is bounded. (iii) Our restriction imposed on α_{n} is much weaker than those in [2, Theorem 2.1].

Furthermore, our result also improves and extends the corresponding results in [1, 3]. For this, we need the following Lemmas.

Lemma 1.1. [1] Let X be real Banach Space and J : X → 2^{X^{∗}} be the normalized duality mapping. Then, for any x, y ∈ X

∥x + y∥^{2} ⩽ ∥x∥^{2} + 2⟨y, j(x + y)⟩, ∀j(x + y) ∈ J(x + y).

Lemma 1.2. [4] Let Φ : [0, ∞) → [0, ∞) be an increasing function with Φ(x) = 0 ⇔ x = 0 and let {b_{n}}_{n=0}^{∞} be a positive real sequence satisfying

∑_{n=0}^{∞} b_{n} = +∞ and lim_{n→∞} b_{n} = 0.

Suppose that {a_{n}}_{n=0}^{∞} is a nonnegative real sequence. If there exists an integer N_{0} > 0 satisfying

a_{n+1}^{2} ⩽ a_{n}^{2} + o(b_{n}) − b_{n}Φ(a_{n+1} ), ∀n ⩾ N_{0},

where lim_{n→∞} \frac{o(b_{n} )}{b_{n}} = 0, then lim_{n→∞} a_{n} = 0.

2. Main results

Theorem 2.1. Let K be a nonempty closed convex subset of a real Banach space X, T : K → K a nearly uniformly L-Lipschitzian (with sequence {r_{n}}_{n⩾0} ) asymptotically generalized Φ−hemicontractive map (with sequence k_{n} ⊂ [1, ∞), lim_{n→∞} k_{n} = 1) such that F (T) = {ρ ∈ K : T ρ = ρ}. Let {α_{n}}_{n⩾0} , {β_{n}^{k}}_{n⩾0} be real sequences in [0, 1] satisfying the following conditions:

(i) ∑ _{n⩾0} α_{n} = ∞

(ii) lim_{n→∞} α_{n} = 0 = β_{n}^{k}, k = 1, 2, …, p − 1.

For arbitrary x_{0} ∈ K, let {x_{n}}_{n⩾0} be iteratively defined by (0.1). Then, {x_{n}}_{n⩾0} converges strongly to ρ ∈ F (T).

Proof. Since T : K → K is an asymptotically generalized Φ-hemicontractive mapping, there exists a strictly increasing continuous function Φ : [0, ∞) → [0, ∞) with Φ(0) = 0 such that

(2.1)  ⟨T^{n}x − T^{n}ρ, j(x − ρ)⟩ ⩽ k_{n}∥x − ρ∥^{2} − Φ(∥x − ρ∥),

for x ∈ K, ρ ∈ F (T), that is

(2.2)  ⟨(T^{n} − k_{n}I)x − (T^{n} − k_{n}I)ρ, j(x − ρ)⟩ ⩽ −Φ(∥x − ρ∥).

Choose an x_{0} ∈ K and x_{0} ≠ T x_{0} such that ∥x_{0}−T^{n}x_{0}∥∥x_{0}−ρ∥+(k_{n}−1)∥x_{0}−ρ∥^{2} ∈ R(Φ) and denote a_{0} = ∥x_{0} − T^{n}x_{0}∥∥x_{0} − ρ∥ + (k_{n} − 1)∥x_{0} − ρ∥^{2}. Indeed, if Φ(a) → +∞ as a → ∞, then a_{0} ∈ R(Φ); if sup{Φ(a) : a ∈ [0, ∞]} = a_{1} < +∞ with a_{1} < a_{0} Then for ρ ∈ K, there exists a sequence {u_{n}} in K such that u_{n} → ρ as n → ∞ with u_{n} ≠ ρ. Clearly, T u_{n} → T ρ as n → ∞ thus {u_{n} −T u_{n}} is a bounded sequence. Therefore, there exists an n_{0} such that ∥u_{n}−T^{n}u_{n}∥∥u_{n}−ρ∥+(k_{n}−1)∥u_{n}−ρ∥^{2} < \frac{a_{1}}{2} for n ⩾ n_{0}. Then we redefine x_{0} = u_{n_{0}} and ∥x_{0}−T^{n}x_{0}∥∥x_{0}−ρ∥+(k_{n}−1)∥x_{0}−ρ∥^{2} ∈ R(Φ). This is to ensure that Φ^{−1} (a_{0}) is well defined.

We first show that {x_{n}}_{n=0}^{∞} is a bounded sequence.

Set R = Φ^{−1} (a_{0}); then from (2.2), we obtain that ∥x_{n} − ρ∥ ⩽ R. Denote
B_{1} = {x ∈ K : ∥x − ρ∥ ⩽ R}, B_{2} = {x ∈ K : ∥x − ρ∥ ⩽ 2R}.

Now, we want to prove that x_{n} ∈ B_{1}. If n = 0, then x_{0} ∈ B_{1}. Now assume that it holds for some n, that is, x_{n} ∈ B_{1}. Suppose that, it is not the case, then ∥x_{n+1} − ρ∥ > R > \frac{R}{2} .

Since {r_{n}} ∈ [0, ∞] with r_{n} → 0. Let M = sup{r_{n} : n ∈ N} and denote

τ_{0} = min\{1, \frac{Φ(R/2)}{24R^{2}} , \frac{Φ(R/2)}{12R[(2R + M)L + R]} , \frac{Φ(R/2)}{12R[2((2R + M)L + R) + M]L} \}.

284 MOGBADEMU

Since lim_{n→∞} α_{n} = 0 = β_{n}^{k}, for k = 1, 2, …, p − 1 and lim_{n→∞} k_{n} = 1. Without loss of generality, let 0 ⩽ α_{n}, β_{n}^{k}, k_{n} − 1 ⩽ τ_{0} for any n ⩾ 0. Then, we have the following estimates from (2.1) for k = 1, 2, …, p − 1.

∥y_{n}^{p−1} − ρ∥ ⩽ (1 − β_{n}^{p−1} )∥x_{n} − ρ∥ + β_{n}^{p−1} ∥T^{n}x_{n} − ρ∥ ⩽ R + τ_{0}L(R + M) ⩽ 2R,

then y^{p−1} ∈ B_{2}. Similarly, ∥y_{n}^{p−2} − ρ∥ ⩽ (1 − β_{n}^{p−2} )∥x_{n} − ρ∥ + β_{n}^{p−2} ∥T^{n}y_{n}^{p−1} − ρ∥ ⩽ R + τ_{0}L(2R + M) ⩽ 2R, then y^{p−2} ∈ B_{2} …, we have

∥y_{n}^{1} − ρ∥ ⩽ (1 − β_{n}^{1})∥x_{n} − ρ∥ + β_{n}^{1}∥T^{n}y_{n}^{2} − ρ∥ ⩽ R + τ_{0}L(2R + M) ⩽ 2R, then y^{1} ∈ B_{2}. Therefore, we get

∥x_{n+1} − ρ∥ ⩽ (1 − α_{n})∥x_{n} − ρ∥ + α_{n}∥T^{n}y_{n}^{1} − ρ∥ ⩽ R + τ_{0}L(2R + M) ⩽ 2R. Also we have the following relations,

∥x_{n+1} − x_{n}∥ ⩽ α_{n}∥T^{n}y_{n}^{1} − x_{n}∥ ⩽ α_{n}(∥T^{n}y_{n}^{1} − ρ∥ + ∥x_{n} − ρ∥)
⩽ τ_{0}(L(2R + M) + R).
∥y_{n}^{1} − x_{n+1} ∥ ⩽ β_{n}∥T^{n}y_{n}^{2} − x_{n}∥ + α_{n}∥T^{n}y_{n}^{1} − x_{n}∥
⩽ β_{n}(∥T^{n}y_{n}^{2} − ρ∥ + ∥x_{n} − ρ∥) + α_{n}(∥T^{n}y_{n}^{1} − ρ∥ + ∥x_{n} − ρ∥)
⩽ 2τ_{0}(L(2R + M) + R).

Using Lemma 1.1 and the above relations, we have

(2.3)  ∥x_{n+1} − ρ∥^{2} ⩽ ∥x_{n} − ρ∥^{2} + 2α_{n}⟨T^{n}y_{n}^{1} − x_{n}, j(x_{n+1} − ρ)⟩

= ∥x_{n} − ρ∥^{2} + 2α_{n}⟨T^{n}x_{n+1} − x_{n+1} , j(x_{n+1} − ρ)⟩ + ⟨x_{n+1} − x_{n}, j(x_{n+1} − ρ)⟩ + ⟨T^{n}y_{n}^{1} − T^{n}x_{n+1} , j(x_{n+1} − ρ)⟩

⩽ ∥x_{n} − ρ∥^{2} + 2α_{n}(k_{n}∥x_{n+1} − ρ∥^{2} − Φ(∥x_{n+1} − ρ∥)) − 2α_{n}∥x_{n+1} − ρ∥^{2} + 2α_{n}L(∥y_{n}^{1} − x_{n+1} ∥)∥x_{n+1} − ρ∥ + 2α_{n}∥x_{n+1} − x_{n}∥∥x_{n+1} − ρ∥ ⩽ ∥x_{n} − ρ∥^{2} + 2α_{n}(k_{n} − 1)∥x_{n+1} − ρ∥^{2}

− 2α_{n}Φ(∥x_{n+1} − ρ∥))

+ 2α_{n}L(∥y_{n}^{1} − x_{n+1} ∥)∥x_{n+1} − ρ∥ + 2α_{n}∥x_{n+1} − x_{n}∥∥x_{n+1} − ρ∥

⩽ ∥x_{n} − ρ∥^{2} − 2α_{n}Φ(R/2) + 2α_{n} \frac{Φ(R/2)}{6}

+ 2α_{n} \frac{Φ(R/2)}{12R} 2R + 2α_{n} \frac{Φ(R/2)}{12R} 2R

⩽ ∥x_{n} − ρ∥^{2} − α_{n}Φ(R/2) ⩽ R^{2}.

which is a contradiction. Hence {x_{n}}_{n=0}^{∞} is a bounded sequence.

We next prove that ∥x_{n} − ρ∥ → 0 as n → ∞. Since lim_{n→∞} α_{n} = 0 = β_{n}^{k}, lim_{n→∞} k_{n} = 1 and {x_{n}} is bounded. Clearly,

lim_{n→∞} ∥x_{n+1} − x_{n}∥ = 0, lim_{n→∞} L∥y_{n}^{1} − x_{n+1} ∥ = 0.

Thus from (2.3), we have

∥x_{n+1} − ρ∥^{2} ⩽ ∥x_{n} − ρ∥^{2} + 2α_{n}⟨T^{n}y_{n}^{1} − x_{n}, j(x_{n+1} − ρ)⟩
= ∥x_{n} − ρ∥^{2} + 2α_{n}⟨T^{n}x_{n+1} − x_{n+1} , j(x_{n+1} − ρ)⟩
+ ⟨x_{n+1} − x_{n}, j(x_{n+1} − ρ)⟩
+ ⟨T^{n}y_{n}^{1} − T^{n}x_{n+1} , j(x_{n+1} − ρ)⟩
⩽ ∥x_{n} − ρ∥^{2} + 2α_{n}(k_{n}∥x_{n+1} − ρ∥^{2} − Φ(∥x_{n+1} − ρ∥))
− 2α_{n}∥x_{n+1} − ρ∥^{2} + 2α_{n}L(∥y_{n}^{1} − x_{n+1} ∥)∥x_{n+1} − ρ∥
+ 2α_{n}∥x_{n+1} − x_{n}∥∥x_{n+1} − ρ∥
⩽ ∥x_{n} − ρ∥^{2} + 2α_{n}(k_{n} − 1)∥x_{n+1} − ρ∥^{2}
− 2α_{n}Φ(∥x_{n+1} − ρ∥)) + 2α_{n}L(∥y_{n}^{1} − x_{n+1} ∥)∥x_{n+1} − ρ∥
+ 2α_{n}∥x_{n+1} − x_{n}∥∥x_{n+1} − ρ∥
= ∥x_{n} − ρ∥^{2} − 2α_{n}Φ(∥x_{n+1} − ρ∥) + o(α_{n}),

where

2α_{n}(k_{n} − 1)∥x_{n+1} − ρ∥^{2} + 2α_{n}L(∥y_{n}^{1} − x_{n+1} ∥)∥x_{n+1} − ρ∥
+2α_{n}∥x_{n+1} − x_{n}∥∥x_{n+1} − ρ∥ = o(α_{n}).

By Lemma 1.2, we obtain lim_{n→∞} ∥x_{n} − ρ∥ = 0. □

Remark 2.1. If we set p = 0 and β_{n}^{1} = 0, then the modified version of the result of [2] holds as a special case of our theorem.

References

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2. J. K. Kim, D. R. Sahu, Y. M. Nam, Convergence theorem for fixed points of nearly uniformly L-Lipschitzian asymptotically generalized Φ-hemicontractive mappings, Nonlinear Anal., Theory Methods Appl. 71(12) (2009), e2833–e2838.

3. A. A. Mogbademu, Z. Xue, Some convergence results for nonlinear maps in Banach spaces, Int. J. Open Problems Compt. Math. 6(1) (2013), 1–10.

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Department of Mathematics (Received 22 07 2013) University of Lagos (Revised 23 09 2014) Lagos, Nigeria

  amogbademu@unilag.edu.ng